Vol. 267, No. 1, 2014

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Normal states of type III factors

Yasuyuki Kawahigashi, Yoshiko Ogata and Erling Størmer

Vol. 267 (2014), No. 1, 131–139
Abstract

Let M be a factor of type III with separable predual and with normal states φ1,,φk,ω with ω faithful. Let A be a finite-dimensional C-subalgebra of M. Then it is shown that there is a unitary operator u M such that φi Adu = ω on A for i = 1,,k. This follows from an embedding result of a finite-dimensional C-algebra with a faithful state into M with finitely many given states. We also give similar embedding results of C-algebras and von Neumann algebras with faithful states into M. Another similar result for a factor of type II1 instead of type III holds.

Dedicated to Masamichi Takesaki on the occasion of his eightieth birthday.

Keywords
von Neumann algebra, type III factor, normal state
Mathematical Subject Classification 2010
Primary: 46L30
Milestones
Received: 27 February 2013
Revised: 30 May 2013
Accepted: 7 June 2013
Published: 22 December 2013
Authors
Yasuyuki Kawahigashi
Graduate School of Mathematical Sciences
The University of Tokyo
Komaba, Tokyo, 153-8914
Japan
Kavli IPMU (WPI)
The University of Tokyo
5-1-5 Kashiwanoha, Kashiwa, 277-8583
Japan
Yoshiko Ogata
Graduate School of Mathematical Sciences
The University of Tokyo
Komaba, Tokyo, 153-8914
Japan
Erling Størmer
Department of Mathematics
University of Oslo
P.O. box 1053
Blindern, NO-0316 Oslo
Norway